Simplify/Condense 3 natural log of x+5 natural log of y-6 natural log of z

$3 \ln (x) + 5 \ln (y) - 6 \ln (z)$

Step 1

Simplify each term.

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$\ln (x^{3}) + 5 \ln (y) - 6 \ln (z)$

Simplify $5 \ln (y)$ by moving $5$ inside the logarithm.

$\ln (x^{3}) + \ln (y^{5}) - 6 \ln (z)$

Simplify $- 6 \ln (z)$ by moving $6$ inside the logarithm.

$\ln (x^{3}) + \ln (y^{5}) - \ln (z^{6})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{3} y^{5}) - \ln (z^{6})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x^{3} y^{5}}{z^{6}})$